Nuclear magnetic resonance (NMR) can be used to determine properties of a substance. Oil and gas field tools use NMR systems to investigate the properties of earth formations, such as the porosity of formations or the composition of fluids within the formations. In one example, the NMR system is lowered into a wellbore that traverses a formation. The NMR system can be lowered into the wellbore as part of a wireline or drilling string. The NMR system is used to investigate the properties of the formation adjacent to the wellbore. The system includes a magnet for applying a static magnetic field to the formation adjacent to the wellbore. The system also includes a coil for applying an oscillating magnetic field to the formation adjacent to the wellbore at a particular frequency. The oscillating field is composed of a sequence of radio frequency (RF) pulses that tip the magnetization of the atomic nuclei within the formation away from an initial magnetization produced by the magnet. The sequence of pulses and the static magnetic field interact with the nuclei in a manner such that a NMR signal composed of “echoes” is generated within at least a portion of the formation. The NMR signal within the formation is detected using the coil and used to determine the properties for the formation.
The static magnetic field produced by the magnet and applied to the formation is inhomogeneous because the field decreases in strength as a function of distance from the magnet. Because of this inhomogeneous field, the NMR signal is produced in a small portion of the formation because the Larmor frequency condition (or resonance condition) is met in a small portion of the field. As is known in the art, the Larmor frequency of the atomic nuclei depends on the strength of the static magnetic field according to the following relationship:ω0=γ×B0 where B0 is the strength of the static magnetic field, γ is the gyromagnetic ratio of the atomic nuclei of interest, and ω0 is the frequency of the resonant signal that is produced by the atomic nuclei (the Larmor frequency). The small portion of the formation (or substance) that produces the resonant signal is referred to as a “shell” or “slice.” In some cases, the shell has a volume of 10−4 m3 (100 cc). These small shell sizes lead to a degradation in the ratio of signal power to noise power (SNR).
The shell sizes and resulting SNR can be increased by increasing the bandwidth of the RF pulses in the NMR sequence. One approach to expand the bandwidth of RF pulses is to increase the amplitude of the RF pulses in the NMR sequence by a factor, while simultaneously decreasing the duration of the pulses by the same factor. The resulting bandwidth and SNR will then increase by this factor. This solution, however, requires increasing RF field strength and average power consumption, which can be problematic for power constrained applications, such as NMR well logging applications.
Another approach is to modulate the amplitude and/or phase within the RF pulses in a particular manner. For example, the bandwidth of RF pulses can be increased by using composite, shaped, chirped, or adiabatic pulses, as compared to conventional rectangular pulses that are designed to have constant amplitude and phase. Progress has been made in the development of numerically-optimized broadband RF pulses that are robust with respect to variation of the RF field strength and other constraints. These developments have taken advantage of the availability of new algorithms for pulse sequence design based on methods of optimal control theory (OCT), which make it feasible to find new pulse sequences in an efficient manner in a high-dimensional parameter space. However, such work has been largely focused on improving the pulse fidelity, i.e., ability to approximate a given rotation operator over the desired bandwidth, and not on SNR. In addition, such pulses are typically much longer than a conventional pulse. This increase in pulse duration has the undesirable consequence that it entails an increase of the minimum echo spacing and an increase in the power consumption per RF pulse, often resulting in no net increase of SNR per unit time. This fact is of particular concern for mobile and field applications, such as NMR well logging applications.